Question

If cosec theta + cot theta is equal to 5 then evaluate sec theta

Answer 1

Answer:

sec theta=13/12

Step-by-step explanation:

cosec theta + cot theta=5......(1)

we know that,

cosec^2theta-cot^2theta=1

it can written as,

(cosec+cot)(cosec-cos) =1

therefore,

cosec theta-cot theta=1/cosec+cos

from this we get,

cosec theta -cot theta=1/5......(2)

add (1)and(2)

cosec theta+cot theta=5....(1)

cosec theta-cot theta=1/5.....(2)

2cosec =5+1/5

=25+1/5

2cosec =26/5

cosec=26/5*1/2

cosec theta=13/5

cosec=1/sin

therefore sin theta=5/13

we know that,

sin^2theta+cos^2theta=1

cos^2theta=1-sin^2theta

=1-(5/13)^2

=1-25/169

=169-25/169

=144/169

cos theta=12/13

sec=1/cos

therefore sec theta =13/12

Answer 2

Answer:

$sec\theta=\frac{13}{12}$

Step-by-step explanation:

In the question,

We know that,

$cosec^{2}\theta - cot^{2}\theta = 1$

So,

$cosec^{2}\theta - cot^{2}\theta = 1\\(cosec\theta +cot\theta)(cosec\theta - cot\theta)=1$

Now,

We have been provided that,

$cosec\theta+cot\theta=5\ (given).........(1)$

So, on putting it in the equation we get,

$cosec^{2}\theta - cot^{2}\theta = 1\\(cosec\theta +cot\theta)(cosec\theta - cot\theta)=1\\5(cosec\theta - cot\theta)=1\\(cosec\theta - cot\theta)=\frac{1}{5}\ ..............(2)$

Now, on adding Equation (1) and (2) we get,

$2cosec\theta=5+\frac{1}{5}=\frac{26}{5}\\cosec\theta=\frac{13}{5}\\So,\\sin\theta=\frac{5}{13}$

Now, we know that,

$sin^{2}\theta +cos^{2}\theta=1\\So,\\On\ putting\ we\ get,\\(\frac{5}{13})^{2}+cos^{2}\theta=1\\cos^{2}\theta=1-\frac{25}{169}\\cos^{2}\theta=\frac{144}{169}\\cos\theta=\frac{12}{13}\\So,\\sec\theta=\frac{13}{12}$

Therefore, we have the value of secθ, which is given by,

$sec\theta=\frac{13}{12}$